3-D TRAVELTIME MAPS

The first example shown in Figure 2 represents sections through a 3-D traveltime field for a constant velocity of 2 km/s. The $100 \times 100 \times 100$ grid is evenly sampled laterally and in depth, with a sample interval of 10 m. The source is situated at the surface, in the middle of the plane. Figure 2a and 2b represent vertical orthogonal planes passing through the source. The vertical plane in Figure 2d is offset with a distance (X=300 m) from the source. Figure 2c shows the values of the traveltime field on a horizontal surface passing through the middle of the cube. Figure 3 displays the difference between the analytical solution and the computed traveltime field for each panel shown in Figure 2. The distribution of the errors is shown through lines of constant value. The errors accumulate with distance from the source. The values of the absolute error on each surface (of order 10^-6) are shown in Figure 4.

 

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Figure 1 Traveltime map for a constant 3-D velocity model (1000 m each side) with the source at the surface. a) Vertical section through the middle of the cube for constant X. b) Vertical section through the middle of the cube for constant Y. c) Horizontal section through the middle of the cube for constant Z. d) Vertical section off the middle of the cube for constant X (X=800 m).

 

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Figure 2 Difference between the computed traveltime map and the analytical solution for a constant velocity model. Each figure represents the difference between the analytical solution and the corresponding panel in Figure 2. The error accumulates with distance from the source.

 

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Figure 3 Errors in the finite-difference calculations. Each panel shows several curves which represent the absolute error across the corresponding sections in Figure 3.

The constant velocity model is a poor example to examine a finite-difference algorithm based on the Engquist-Osher scheme, as the functions $u(r,\theta,\phi)$ and $v(r,\theta,\phi)$ are zero on spherical shells of constant radius r. Therefore the errors in the constant velocity case are merely due to numerical integration and interpolation from cartesian to spherical coordinates.

Figure 5 shows results for a traveltime field in a depth variable velocity medium. the computations are done on the same $100 \times 100 \times 100$grid with a spacing of 10 m. The three panels shown in Figures 5a, 5b and 5c represent vertical planes situated at increasing distance from the source. Figure 5d displays the depth variable velocity model. The 3-D algorithm is an order of magnitude more unstable than the 2-D algorithm for comparable grid parameters in spherical, and cylindrical coordinates respectively. The instability problems disappear for smaller grid intervals in spherical coordinates, but one is limited by the amount of computer memory which can be allocated for an increasing number of grid points. The absolute error is not negligible at the sides of the panel, as can be seen from Figure 6c and 6d. Maximum values for the error appear on the edges of the model studied, at higher values of the traveltime field. The traveltimes in the 3-D model tend to be higher, as the difference between the 3-D section and the 2-D section is always positive.

 

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Figure 4 Traveltime map for a 3-D depth variable velocity model (1000 m each side) with the source in the middle of the top surface. a) Vertical section through the source. b) Vertical section with offset (X=250 m) from the source. c) Vertical section with offset (X=450 m) from the source. d) Velocity model.

 

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Figure 5 Comparison between 3-D and a 2-D finite-difference traveltime map for the depth variable velocity model shown in Figure 5d. a) Vertical section through the source in the 3-D model. b) 2-D traveltime field for the same depth variable velocity model. c) Difference between the 3-D traveltime field and the 2-D traveltime field. d) Absolute errors for the 3-D vs. 2-D difference.